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Mirrors > Home > MPE Home > Th. List > gictr | Structured version Visualization version GIF version |
Description: Isomorphism is transitive. (Contributed by Mario Carneiro, 21-Apr-2016.) |
Ref | Expression |
---|---|
gictr | ⊢ ((𝑅 ≃𝑔 𝑆 ∧ 𝑆 ≃𝑔 𝑇) → 𝑅 ≃𝑔 𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brgic 19060 | . 2 ⊢ (𝑅 ≃𝑔 𝑆 ↔ (𝑅 GrpIso 𝑆) ≠ ∅) | |
2 | brgic 19060 | . 2 ⊢ (𝑆 ≃𝑔 𝑇 ↔ (𝑆 GrpIso 𝑇) ≠ ∅) | |
3 | n0 4307 | . . 3 ⊢ ((𝑅 GrpIso 𝑆) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝑅 GrpIso 𝑆)) | |
4 | n0 4307 | . . 3 ⊢ ((𝑆 GrpIso 𝑇) ≠ ∅ ↔ ∃𝑔 𝑔 ∈ (𝑆 GrpIso 𝑇)) | |
5 | exdistrv 1960 | . . . 4 ⊢ (∃𝑓∃𝑔(𝑓 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑔 ∈ (𝑆 GrpIso 𝑇)) ↔ (∃𝑓 𝑓 ∈ (𝑅 GrpIso 𝑆) ∧ ∃𝑔 𝑔 ∈ (𝑆 GrpIso 𝑇))) | |
6 | gimco 19059 | . . . . . . 7 ⊢ ((𝑔 ∈ (𝑆 GrpIso 𝑇) ∧ 𝑓 ∈ (𝑅 GrpIso 𝑆)) → (𝑔 ∘ 𝑓) ∈ (𝑅 GrpIso 𝑇)) | |
7 | brgici 19061 | . . . . . . 7 ⊢ ((𝑔 ∘ 𝑓) ∈ (𝑅 GrpIso 𝑇) → 𝑅 ≃𝑔 𝑇) | |
8 | 6, 7 | syl 17 | . . . . . 6 ⊢ ((𝑔 ∈ (𝑆 GrpIso 𝑇) ∧ 𝑓 ∈ (𝑅 GrpIso 𝑆)) → 𝑅 ≃𝑔 𝑇) |
9 | 8 | ancoms 460 | . . . . 5 ⊢ ((𝑓 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑔 ∈ (𝑆 GrpIso 𝑇)) → 𝑅 ≃𝑔 𝑇) |
10 | 9 | exlimivv 1936 | . . . 4 ⊢ (∃𝑓∃𝑔(𝑓 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑔 ∈ (𝑆 GrpIso 𝑇)) → 𝑅 ≃𝑔 𝑇) |
11 | 5, 10 | sylbir 234 | . . 3 ⊢ ((∃𝑓 𝑓 ∈ (𝑅 GrpIso 𝑆) ∧ ∃𝑔 𝑔 ∈ (𝑆 GrpIso 𝑇)) → 𝑅 ≃𝑔 𝑇) |
12 | 3, 4, 11 | syl2anb 599 | . 2 ⊢ (((𝑅 GrpIso 𝑆) ≠ ∅ ∧ (𝑆 GrpIso 𝑇) ≠ ∅) → 𝑅 ≃𝑔 𝑇) |
13 | 1, 2, 12 | syl2anb 599 | 1 ⊢ ((𝑅 ≃𝑔 𝑆 ∧ 𝑆 ≃𝑔 𝑇) → 𝑅 ≃𝑔 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∃wex 1782 ∈ wcel 2107 ≠ wne 2944 ∅c0 4283 class class class wbr 5106 ∘ ccom 5638 (class class class)co 7358 GrpIso cgim 19048 ≃𝑔 cgic 19049 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-1o 8413 df-map 8768 df-0g 17324 df-mgm 18498 df-sgrp 18547 df-mnd 18558 df-mhm 18602 df-grp 18752 df-ghm 19007 df-gim 19050 df-gic 19051 |
This theorem is referenced by: gicer 19067 cyggic 20982 |
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